Thursday, March 21, 2019

DM 41L: A Song of Irrational Numbers

Tones of the first ten digits of the constants π, √2, Zeta(2), Phi (Golden Ratio constant), and e (Euclidean constant) using the Swiss Micros DM 41L


Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Tuesday, March 19, 2019

Algebra: Solving Simple Non-Linear Systems

Algebra: Solving Simple Non-Linear Systems




System I:  

x + y = a
x^2 + y = b

Solving for y:
x + y = a
y = a - x

Subtracting the two equations from the system:
x + y = a
- [x^2 + y] = -[ b ]

x - x^2 = a - b
x^2 - x = b - a
x^2 - x - (b - a) = 0

Solving for x:
x = ( 1 ± √(1 - 4*(b - a) ) / 2

Summary for System I:
x = ( 1 ± √(1 - 4*(b - a) ) / 2
y = a - x

If a and b are real numbers, then 1 - 4*(b - a) ≥ 0, and
1 ≥ 4*(b - a)

System II:

x + y = a
x + y^2 = b

Solving for x:
x + y = a
x  = a - y

Subtracting the two equations from the system:
x + y = a
- [ x + y^2 ] = -[ b ]

y - y^2 = a - b
y^2 - y = b - a
y^2 - y - (b - a) = 0

Solving for y:
y = ( 1 ± √(1 - 4*(b - a) )/2

Summary for System II:
x  = a - y
y = ( 1 ± √(1 - 4*(b - a) )/2

System III:

x + y = a
x^2 + y^2 = b

Solving for y:
y = a - x

Solving for x:
x^2 + (a - x)^2 = b
x^2 + a^2 - 2*a*x + x^2 = b
2*x^2 - 2*a*x + (a^2 - b) = 0

x = ( 2*a ± √(4*a^2 - 4*2*(a^2 - b) ) / 4
x = ( 2*a ± √(4*a^2 - 8*(a^2 - b) ) / 4
x = ( 2*a ± √(4*a^2 - 8*a^2 + 8*b) ) / 4
x = ( 2*a ± √(8*b - 4*a^2) ) / 4
x = ( a ± √(2*b - a^2) ) / 2

Summary for System III:
x = ( a ± √(2*b - a^2) ) / 2
y = a - x

System IV:

x^2 + y^2 = a
x * y = b

Solving for y:
y = b / x 

I'm assuming that x ≠0 and y ≠0.

x^2 + y^2 = a
x^2 + (b / x)^2 = a
x^4 + b^2 = a * x^2
x^2 - a * x^2 + b^2 = 0

Let w = x^2, then w^2 = x^4

Then:
w^2 - a*w + b^2 = 0

Then:
w = (a ± √(a^2 - 4 * b^2) )/ 2

And:
x = ± √( (a ± √(a^2 - 4 * b^2) )/ 2 )

We have four answers to the system.

Summary for System IV:
x = ± √( (a ± √(a^2 - 4 * b^2) )/ 2 )
y = b / x 

A lot of fun,

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Thursday, March 14, 2019

Birthday Blog: Fun Facts About Me

Fun Facts About, me, Eddie Shore, the author of Eddie's Math and Calculator Blog:





1.  I have a Bachelor's Degree in Accounting and a Master's Degree in Mathematics, both from Cal Poly Pomona.

2.  I don't have a favorite sport.

3. My favorite video games are Super Mario Maker, Super Mario Brothers, Mario Kart, Millipede, and Joust.

4. My favorite colors are sky blue, denim blue, forest green, and gold.

5. For you zodiac fans, my zodiac sign is Pisces.  Both of my parents are Geminis, and most of my closest friends are Scorpios.

6. Chocolate chip cookies are my weakness.

7. My four favorite go-to music artists are Earth, Wind & Fire, Stevie Wonder, Janet Jackson, and Sheryl Crow.

8. At home, I have two dogs and three cats. 

9.  I'm Irish and Mexican.

10.  Every morning I write what I want to accomplish for the day.  I go through a lot of Post-Its.

11. The beach is my sanctuary.

12. The mathematics section of university library is to me what Disneyland is to most people, only I don't have to pay $60 for an admission ticket. ;)  My favorite library is the Honnold Mudd Library in Claremont (Claremont Colleges).

13. I can't live without music.  Or calculators.

14. My dream car is a Ferrari. 

15. I prefer tea over coffee.

16. My turn ons are honesty, warmth, kindness, intelligence, and a love for life.

17. My turn offs are dishonesty, arrogance, racism, sexism, and ageism.

18. My bucket list grows by the day.

19. I am very close to my family. 

20. My favorite vacation spot so far is Maui.

21. My favorite number is pi (π), partly because my birthday is March 14 (the day that this blog is posted).

22. I want to go to Greece, Italy, Ireland, and New Zealand.  I'd probably wouldn't return home. 

23. My guilty pleasure is the Real Housewives of New Jersey.  #TeamMargaretJosephs

24. I'm nearsighted and prefer glasses to contacts.

25. I prefer wine over beer, but I do enjoy a good ale.  My favorite shot is Fireball.

26. My newest favorite YouTube channel is Doctor Mike. Favorite of all time is both Cinemasins and Music Video Sins.

27. My favorite calculators are the HP Prime, HP 42S, HP 12C, TI-84 Plus CE and Casio fx-991EX.

28. I am a Press Your Luck addict. Big Bucks, no whammies!

29. I have two dogs and three cats.

30. My music tastes are kind of eccentric, but I learn towards rock, alternative, and some R & B.  I do have a Spotify account. 

31.  My favorite fruits of cherries, applies, and strawberries.

32.  My favorite holiday used to be Christmas, now it's Halloween.

33.  A hobby I like but don't do enough of is art.  I'm drawn to glass art, mythological art, and fantasy art. 

34.  My favorite movie is Ghostbusters - the 1984 original one.

35.  My favorite pie is apple. 

36.  My favorite fonts are Arial, Courier, Futura, and Banschrift.

37.  I lived in Southern California all my life.

38.  Starting and writing on this blog is one of the most fun things I am very fortunate to do, and I thank you for reading and supporting this blog. 






Eddie
All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.




Wednesday, March 13, 2019

TI-84 Plus: Greenwich Mean Sidereal Time Estimate

TI-84 Plus:  Greenwich Mean Sidereal Time Estimate

Introduction

The program GMST calculates and estimates the Greenwich Mean Sidereal Time for any dates between January 1, 1900 and December 31, 2099.

For any date between 1900 and 2099, the date number is calculated as:

number of days since January 1 + (year - 1900) * 365 + int((1900 - year)/4) + 0.5 + hour/24

Please keep in mind, the formula in this program is from 1978 (see source).

For the hour, a 2400 hour clock format is used.  For example, 1 AM = 1, 1 PM = 13. 

This program does not take the location of the observer into account.

TI-84 Plus Program: GMST

"2019-03-08 EWS"
Disp "1900-2099"
Input "MONTH: ",M
Input "DAY: ",D
Input "YEAR: ",Y
Input "HOUR: ",H
If fPart(Y/4)=0 and Y≠1900
Then
1→L
Else
0→L
End

If M≥3
Then
int(30.6*M+1.6)+D-35+L→T
Else
int(30.6*M+368.8)+D-400→T
End

T+(Y-1900)*365+iPart((Y-1900)/4)+.5+H/4Z
Z/36525→Z
6°38'45.836"+2400.051262*Z+0°0'0.0929"*Z²→E
24*fPart(E/24)→E
Disp "GMST: ",E>DMS


Example:

Example 1: 

January 1, 1978, Midnight (Hour = 0):  6°41'9.836"

Example 2:

March 13, 2011, 7:00 PM (H = 19): 11°23'54.646"

Source:

Jones, Aubrey  Mathematical Astronomy With a Pocket Calculator  Halsted Press:  John Wiley & Sons, New York.  1978.  ISBN 0 470 26552 3

Eddie


All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Sunday, March 10, 2019

TI-84 Plus and HP 41C: Number of Days After January 1

TI-84 Plus and HP 41C:  Number of Days After January 1

Introduction

The program DATENO calculates the number of days from January 1.  The program prompts whether we are working in a leap year or not. 

With D = Day and M = Month, the days between January 1 and any other date within the calendar year is:

If M = 1 and M = 2 Then
DATE# = int(30.6 * M + 368.8) + D - 400

Otherwise,
DATE# = int(30.6 * M + 1.6) +D - 35  (non-leap year)
DATE# = int(30.6 * M + 1.6) + D - 34  (leap year)

TI-84 Plus Program: DATENO

"DAYS AFTER JANUARY 1"
"2019-03-07 EWS"
Input "MONTH: ",M
Input "DAY: ",D
Disp "0:NO, 1:YES"
Input "LEAP YEAR? ",L
If M≥3
Then
int(30.6*M+1.6)+D-35+L→T
Else
int(30.6*M+368.8)+D-400→T
End
Disp T

HP 41C/DM 41L Program:  DATENO

(^T:  beginning of an alpha string)

01 LBL^T DATENO
02 ^T MONTH
03 PROMPT
04 STO 01
05 ^T DAY?
06 PROMPT
07 STO 02
08 ^T LEAP? N=0,L=1
09 PROMPT
10 STO 03
11 RCL 01
12 3
13 X<=Y?
14 GTO 00
15 30.6
16 RCL 01
17 *
18 368.8
19 +
20 INT
21 RCL 02
22 +
23 400
24 -
25 GTO 01
26 LBL 00
27 RCL 01
28 30.6
29 *
30 1.6
31 +
32 INT
33 RCL 02
34 + 
35 35
36 - 
37 RCL 03
38 + 
39 LBL 01
40 STO 04
41 END

Examples

Days between January 1 and February 16  (M = 2, D = 16):  46

Days between January 1 and October 1 (M = 10, D = 1):
(Non-leap year):  273
(Leap year): 274

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Thursday, March 7, 2019

Google Celebrates Olga Ladyzhenskaya

Google Celebrates Olga Ladyzhenskaya 



Today, March 7, 2019, Google honored the Russian mathematician Olga Ladyzhenskaya.  Ladyzhenskaya was born on March 7, 1922 (passed away on January 12, 2004).  She is known for her work in partial differential equations, particularly providing a rigorous proof of the finite difference method for the Navier-Strokes equations. 

Happy Birthday Olga! And thank you for your contributions to mathematics and science.

Wikipedia articles:

Bio on Olga Ladyzhenskaya
https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya

Navier-Strokes Equations
https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Finite Difference Method
https://en.wikipedia.org/wiki/Finite_difference_method

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.



Wednesday, March 6, 2019

HP 12C and HP 11C: Loan Amount Using the Annual Loan Constant

HP 12C and HP 11C:  Loan Amount Using the Annual Loan Constant

Introduction

The program calculates the theoretical loan amount using the following factors:

*  NOI:  Net Operating Income. The estimated net operating income the property is expected to earn annually.  An average is usually used.

*  DCR:  Debt Coverage Ratio.  The ratio of net operating income to annual debt service, describing a company's ability to pay its debts.  Generally, the larger the DCR, the better.  We really don't want DCR to be below 1.

*  Number of payments per year, number of years, and annual interest rate of the potential loan. 

The ALC, or the annual loan constant is calculated by:

*  Either divided the annual debt service by the loan amount (when the amount is known), or

*  Determining the periodic payment to amortize a $100 loan given number of payments and interest rate.

Set up:
Number of payments -> N
Interest Rate -> I%YR  (or periodic interest rate -> i)
-100 -> PV
0 -> FV
Solve for PMT

The ALC is expressed as a percentage. 

The theoretical loan amount is calculated by:

Loan = NOI / (DCR * ALC%)

HP 12C Program: Loan Amount Using the Annual Loan Constant

Instructions:
Store the following:
NOI in R1
DCR in R2
Number of payments per year in R3
Number of periods in [ n ]
Periodic Interest rate in [ i ]

Program:
Step;  Key;  Code
01;  1;  1
02;  0;  0
03;  0;  0
04;  CHS;  16
05;  PV;  13
06;  0;  0
07;  FV;  15
08;  PMT;  14
09;  RCL 3;  45, 3
10;  *;  20
11;  RCL 2; 45, 2
12;  x<>y;  34
13;  %;  25
14;  RCL 1; 45, 1
15;  x<>y;  34
16;  ÷;  10
17;  GTO 00;  43, 33, 00

(* HP 12C Platinum, step 17:  GTO  000; 43, 33, 000)

HP 11C Program:    Loan Amount Using the Annual Loan Constant

Instructions:
Store the following:
NOI in R1
DCR in R2
Number of payments per year in R3
Number of periods in R4
Periodic Interest rate in R5

Program:
Step; Key; Code
001;  LBL A; 42, 21, 11
002;  1;  1
003;  ENTER; 36
004;  ENTER; 36
005;  RCL 5;  45, 5
006; %;  43, 14
007;  +;  40
008;  RCL 4;  45, 4
009;  CHS;  16
010;  y^x; 14
011;  *;  30
012;  1;  1
013;  RCL 5; 45, 5
014;  %;  43, 14
015;  x<>y;  34
016;  R↓;  33
017;  ÷;  10
018;  1;  1
019;  0;  0
020;  0;  0
021;  x<>y; 34
022;  ÷;  10
023;  RCL 3; 45, 3
024;  *;  20
025;  RCL 2; 45, 2
026;  x<>y; 34
027;  %;  43, 14
028;  RCL 1; 45, 1
029;  x<>y;  34
030;  ÷; 10
031;  RTN; 43, 32

Examples

Example 1: 

NOI:  $58,000.00
DCR:  1.25
P/Y:  12
Number of Years: 30
Annual Interest Rate:  5%

Loan Amount:  $720,288.92

Example 2:

NOI:  $40,000.00
DCR:  1.35
P/Y:  12
Number of Years: 20
Annual Interest Rate:  6.8%

Loan Amount: $323,464.95

Source: 

Goldman, Mark H. and Stephen D. Messner "HP 12C Real Estate Applications Handbook"  Hewlett Packard Rev. B. March 1984

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

DM 41L: A Song of Irrational Numbers

Tones of the first ten digits of the constants π, √2, Zeta(2), Phi (Golden Ratio constant), and e (Euclidean constant) using the Swiss Micr...